“A Whisky Puzzle”

In 1895 a London shopkeeper attracted customers with a glass cask of whiskey — they were puzzled to find that no matter how much liquid they drew off, the level in the cask never dropped. The container could be viewed from any angle, and it stood well away from the wall. How was this possible?

A hidden pipe connected the bottom of the cask to a tank in another room. When a customer drew a glass of whisky, a confederate there would open a tap to replenish tank B, and the liquid, seeking its own level, would maintain the same height in the cask.

(James Scott, “Shopkeepers’ Advertising Novelties,” Strand, November 1895. See Desert Downpour.)

April 22, 2024April 21, 2024 | Puzzles · Science & Math

Constitutional Crisis

From Lee Sallows:

(Thanks, Lee!)

April 17, 2024April 16, 2024 | Puzzles · Science & Math

Black and White

crypton chess puzzle

An old puzzle by Paul Hoffman from Science Digest. Dr. Crypton is playing chess with his boss. Crypton has the white pieces. What move can he play that will not checkmate Black? There’s no funny business; the problem is just what it seems, except that Crypton has promised never to put a knight on any square adjacent to the black king, so 1. Ne6 doesn’t count as a solution.

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1. Rc5+ unpins the black rook on b4, permitting the reply 1. … Rxf4 and thus avoiding mate. From Dr. Crypton and His Problems: Mind Benders From Science Digest, 1982.

April 16, 2024 | Puzzles

Empty Vessel

A puzzle by Soviet science writer Yakov Perelman: A bottle full of gasoline has a mass of 1,000 grams. The same bottle filled with acid has a mass of 1,600 grams. The acid is twice as dense as the gasoline. What’s the mass of the bottle?

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The mass of the bottle plus the mass of the gasoline is 1,000 grams. And the mass of the bottle plus twice the mass of the gasoline is 1,600 grams (because the acid is twice as dense as the gasoline). So the difference, 600 grams, is the mass of the bottle’s volume of gasoline. The mass of the bottle full of gasoline is 1,000 grams, so the mass of the bottle alone is 400 grams. (This checks out: It tells us that the mass of the acid is 1,600 – 400 = 1,200 grams, which is twice the mass of the gasoline.) From the March-April 1993 issue of Quantum (page 32, “Kaleidoscope”).

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April 9, 2024April 8, 2024 | Puzzles

Win Count

Imagine a game of tic-tac-toe (noughts and crosses) played in three dimensions in an 8×8×8 cube. A player wins by marking some straight line of eight cells through the large cube. How many such winning lines are there?

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This problem was proposed in the American Mathematical Monthly by A.L. Rubinoff of the University of Toronto in 1948 (Problem E773). It’s possible simply to count up the lines, but Leo Moser of the University of Manitoba hit on a simpler solution. Encase the 8×8×8 cube in a “shell” of unit thickness, and extend each winning line so that it pierces the shell at either end. Each winning line pierces two cells in the shell, and each cell in the shell is pierced by only one winning line. Thus each winning line corresponds to a unique pair of cells in the shell, and the number of winning lines is just half the number of cells in the shell: $\displaystyle \frac{10^{3} - 8^{3}}{2} = \frac{1000 - 512}{2} = 244.$ This solving strategy can be generalized to cubes of any size. (Via Ross Honsberger, Mathematical Morsels, 1978.)

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April 8, 2024April 7, 2024 | Puzzles

“Three Threes Are Ten”

This little trick often puzzles many:–

Place three matches, coins, or other articles on the table, and by picking each one up and placing it back three times, counting each time to finish with number 10, instead of 9. Pick up the first match and return it to the table saying 1; the same with the second and third, saying 2 and 3; repeat this counting 4; but the fifth match must be held in the hand, saying at the time it is picked up, 5; the other two are also picked up and held in hand, making 6 and 7; the three matches are then returned to the table as 8, 9, and 10. If done quickly few are able to see through it.

— John Scott, The Puzzle King, 1899

04/20/2024 Reader Vladamir Tsepis adds, “This reminds me of the way to convince children you have 11 fingers. Start by showing your left hand splayed, curl down the thumb and index finger counting ‘one, two…’, then of the remaining say ‘let’s skip these three’. Move to your right hand, bend each finger in turn as you count ‘four, five, six, seven, eight…’. Return to the left hand counting off the three we skipped ‘nine, ten, eleven.'”

April 5, 2024April 20, 2024 | Puzzles

Memorial

From The Book of 500 Curious Puzzles, 1859:

Following is the epitaph of Ellinor Bachellor, an old pie woman. How should we read it?

Bene A. Thin Thed Ustt HEMO. Uld yo
L.D.C. RUSTO! Fnel L.B.
Ach El Lor. Lat. ELY,
Wa. S. shove N. W. How — Ass! kill’d I. N. T. H.
Ear T. Sofp, I, Escu Star.
D. San D T Art. San D K. N E. W. E
Ver — Yus E. — Oft He ove N, W. Hens He
‘Dli V’DL. on geno
Ug H S hem A.D.E. he R. la Stp. Uf — fap
Uf. F. B Y he. R hu
S. Ban D. M.
Uch pra is ‘D. No. Wheres Hedot
HL. i. e. Tom. A kead I.R.T.P. Yein hop Esthathe
R. C. RUSTWI,
L L B. Era is ‘–D!

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Beneath in the dust The mouldy old crust Of Nell Bachellor lately was shoven: Who was skilled in the arts Of pies, custards, and tarts, And knew every use of the oven. When she’d liv’d long enough, She made her last puff, A puff by her husband much prais’d: Now here she doth lie, To make a dirt pie, In hopes that her crust will be rais’d.

April 4, 2024April 3, 2024 | Puzzles

Counting Up

A problem from Daniel J. Velleman and Stan Wagon’s excellent 2020 book Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles:

A square grid measures 999×999. Each square is either black or white. Each black square that’s not on the border of the grid has exactly five white squares among its eight immediate neighbors (those that adjoin it horizontally, vertically, or diagonally). Each white square that’s not on the border has exactly four black squares among its immediate neighbors. Of the 999 × 999 = 998001 squares in the grid, how many are black and how many white?

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999 is 3 × 333, so we can divide the grid into 333² 3×3 subgrids. Each of those subgrids consists of a central square surrounded by eight neighbors. According to our information, if a subgrid’s central square is black then exactly five of its neighbors are white, and thus the subgrid contains four black and five white squares. If the central square is white, then the subgrid likewise has four black and five white squares. So the entire grid has 333² × 4 = 443556 black and 333² × 5 = 554445 white squares.

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March 26, 2024March 25, 2024 | Puzzles

Specialist

A puzzle by Soviet science writer Yakov Perelman: Six carpenters and a cabinetmaker were hired to do a job. Each carpenter was paid 20 rubles, and the cabinetmaker was paid 3 rubles more than the average wage of the whole group. How much did the cabinetmaker make?

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We’re told that the cabinetmaker made 3 rubles more than the average wage of the seven together. So if we relieve him of 3 rubles he’ll be left with the average wage. What is that wage? Distribute the 3 rubles we’ve just taken from the cabinetmaker equally among the other six workers. Each will get half a ruble. So the average wage is 20.5 rubles, and the cabinetmaker got 23.5 rubles. From Perelman’s 1924 book For Young Mathematicians, via the March-April 1993 issue of Quantum (“Kaleidoscope,” page 32).

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March 25, 2024March 24, 2024 | Puzzles

Mass Transit

A problem from the October 1964 issue of Eureka, the journal of the Cambridge University Mathematical Society:

The planet Kophikkup is in the shape of a torus or ring-doughnut. There is a direct mono-rail line from each of the four space-ports to each of the major cities. No lines join or cross. What is the greatest possible number of major cities? Draw a diagram for this case.

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Four. The map above shows that each of four spaceports can communicate directly with each of four cities on the surface of the torus, but there’s nowhere to “put” a fifth city that wouldn’t require some of the monorail lines to cross. (From the Eureka Digital Archive, published under a Creative Commons license.) 03/26/2024 Reader Drake Thomas sent a better diagram: (Thanks, Drake.)

March 22, 2024March 26, 2024 | Puzzles